identify the graph of $y = e^{x} - 2$.

Explanation:

Step1: Analyze parent function

The parent function is $y=e^x$, which is an increasing exponential curve with a horizontal asymptote at $y=0$, passing through $(0,1)$.

Step2: Apply vertical transformation

The given function is $y=e^x - 2$, which is a vertical shift down by 2 units of $y=e^x$.

• New horizontal asymptote: $y=0-2=-2$
• New y-intercept: When $x=0$, $y=e^0 - 2=1-2=-1$, so the curve passes through $(0,-1)$.

Step3: Match to graphs

The first graph has asymptote $y=-2$, passes through $(0,-1)$, and is increasing, matching the transformed function. The second graph has asymptote $y=0$ and y-intercept $(0,0)$ (shifted incorrectly), the third is a decreasing exponential (matches $y=e^{-x}$ type, not our function).

Answer:

The first graph (leftmost square with the curve approaching $y=-2$ and passing near $(0,-1)$)